When do I use connaître and when do I use savoir?

A basic rule of thumb often tought is that savoir is used with facts and
connaître is used with people/places.

I'd like to propose a slightly revised version of this rule of thumb:

If you're talking about knowing a person/place, as in 'I know Paul', 'I know London',
then use connaître.
More specifically, these are cases where you can't easily re-formulate your sentence with a
'question-inside-a-sentence' (see below).

In other cases, you can generally use either verb.

By 'question-inside-a-sentence', I'm talking about the difference between a verb phrase and a
noun phrase. In many sentences, it is possible to express a similar meaning with either a verb
phrase or a noun phrase:

Contrast these with sentences such as I don't know London, do you know Paul.'
It's not obvious what verb phrase (or 'question-in-a-sentence') you'd replace London
or Paul with to get a similar meaning.

Coming back to French, an essential difference between the two verbs is that:

The verb savoir can be followed by a verb phrase, whereas
connaître cannot.

So what does that mean? Well, it concerns sentences where savoir can be used
when you have a 'question inside a senetnece':

The underlined phrases in French are noun phrases and don't contain a verb. (But purely as a matter
of translation, you'll see from these examples that it's common to translate them using a
verb in English.)

Note that these sentences are referring to pretty much 'the same facts': the difference in
verb is grammatical rather than to do with whether or not the object is a fact or place.